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spar or other crystal, cut for shewing rings, followed by a Nicol's i                        prism. The plane-polarized pencils respectively stopped by and
1                        transmitted through the Nicol's prism consisted, on entering the
crystal, of pencils elliptically polarized in opposite ways; and i                        the nature of this elliptic polarization changes in every possible
i '                      manner from one point to another of the field of view. If these
i                      two streams of light be equivalent according to the definition
given in  the preceding  article,  they will  present  exactly  the
same appearance on being viewed through a crystal followed by
a Nicol's prism or other analyzer.
11.   THEOREM.    Let a polarized stream be resolved into two 1   j                           oppositely polarized  streams; let the phase of vibration of one
I  M ;                          of the streams be altered by a given quantity relatively to that
I1  j i                          of the other, and let the streams be then compounded.    If the I : If                         polarization of the original stream be now changed to its oppo-I   i)                         site, the polarization of the final stream will also be changed to 1   I! ,                       its opposite.
*f   11 i                             The straightforward mode of demonstrating this theorem, by
!f   !                           making use of the general expressions, would lead to laborious
f   \                           analytical   processes,  which   are  wholly   unnecessary.     For   the
l|   *'                         formulae which determine the components of a given stream are
|    I                         expressed by simple equations, so that the results are unique, and
'/,                             accordingly whenever we can foresee what the result will be, it is
|    ,                         sufficient to shew that the formulas themselves, or the geometrical
j * (!                         conditions of which the formulse are merely the expressions, are
. ( ,                         satisfied.
1 i'                                 For shortness' sake call the original stream X, and its com-
, \'!                         ponents 0, E.    Let p be the given quantity, positive or nega-
tive, by which the phase of vibration of 0 is retarded relatively to that of E.    Let o, e denote the streams 0, E after the changes of phase, and Y the stream resulting from their reunion.    Con-,                          ceive now all the vibrations with which we are concerned to be
turned in azimuth through 90. This will not affect the geometrical relations connecting components and resultants. Let X', 0', E', o'', e, Y be the streams which X, 0, E, o, e, Y thus become. The streams 0'', E' are evidently polarized in the same manner respectively as E, 0, except that right-handed is changed into left-handed, and vice versa; and in passing from 0' to c/ the ;                       phase is retarded by p. Now conceive the direction of motion of